?For a fish swimming at a speed v relative to the water, the energy expenditure per unit
Chapter 4, Problem 52(choose chapter or problem)
For a fish swimming at a speed v relative to the water, the energy expenditure per unit time is proportional to 𝑣3. It is believed that migrating fish try to minimize the total energy required to swim a fixed distance. If the fish are swimming against a current 𝑢 (𝑢 ﹤ 𝑣), then the time required to swim a distance 𝐿 is 𝐿/(𝑣 ⎼ 𝑢) and the total energy 𝐸 required to swim the distance is given by
where 𝑎 is the proportionality constant.
(a) Determine the value of v that minimizes 𝐸.
(b) Sketch the graph of 𝐸.
Note: This result has been verified experimentally; migrating fish swim against a current at a speed 50% greater than the current speed.
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